Thread: Area of a Circular Segment

While working on a problem, I had to look up a formula for the area of a circle segment. I have noticed that there is some ambiguity in the results.
According to Wolfram Alpha, the formula is:


According to two other sources, the formula is:
½r^2(θ × pi/180)

Wolfram Alpha has been very reliable in the past, however I would prefer to not get this wrong due to a formula error. Please clarify which or what is correct. Thanks.

Originally Posted by ATC

While working on a problem, I had to look up a formula for the area of a circle segment. I have noticed that there is some ambiguity in the results.
According to Wolfram Alpha, the formula is:


This is correct for angles expressed in radians.
The triangle from the centre of the circle to the 2 points on the circumference
is subtracted from the sector to obtain the segment.

According to two other sources, the formula is:
½r^2(θ × pi/180)

This is the area of the sector for the angle expressed in degrees.
The formula is converting degrees to radians and calculating the sector area.
The triangle still needs to be subtracted to obtain the segment area.

Wolfram Alpha has been very reliable in the past, however I would prefer to not get this wrong due to a formula error. Please clarify which or what is correct. Thanks.

One formula calculates segment area, the second one calculates sector area.

You can learn the difference in circular segment and circular sector.
Circular Segment -- from Wolfram MathWorld

Circular Sector -- from Wolfram MathWorld